"Real" is never used as an adverb. It is an adjective. If you want an adverb, 'really' is the word you want. This is a real mess. [correct] This is real messy. [incorrect] This is really messy. [correct]

With all due respect to correct English, people use "real" as an adverb all the time, particularly in spoken English, so the asker's question is valid. So, you were basically right. "Real" used as an adverb does mean "very." When "real" is used as an adjective, it is describing something of reality, something genuine or not make-believe. Except if it's used in an example like "This is a real mess." In that case, the meaning is really to emphasize the extent of the mess. Having said that, I don't recommend that you or other students use "real" as an adverb, because it's wrong. But it's good to ask questions about it, because we still have to understand the language as we see and hear it used. Just know that it is grammatically wrong to use "real" as an adverb.

Yes, you are right. But I guess adverb "really" is more often used to mean "actually".